Etydhv

I am teaching a course in history of mathematics and would like to draw something like the following:

The bullet points should be possible to draw on the lines and in between. Can anyone help me? Thank you very much!

As you are going to use this quite a bit you'll need a reasonable interface, so how about using code like this

 abacus2,1,1,1,3,3,2,,4,2,1
 qquad
 abacus1,,,,2,1

to produce something like this:

The way that this works is that the abacus command accepts a comma separated list of "abacus rows", which start from the bottom of the abacus and climb upwards. Each "abacus row" is itself a comma separated list that gives the number of balls in each column of the abacus, from left to right.

Here's the full code:

documentclassarticle
usepackagetikz

tikzset
 pics/abacus/.style = 
 code=
 ifnum#1>0% have to treat 0 balls separately
 foreach ball [evaluate=ball as x 
 using (ball-0.5-#1/2)*0.2] in 1,...,#1 
 fill[black] (x,0) circle (2.5pt);
 
 fi
 
 
 ,
 /tikz/abacus/.is family,
 /tikz/abacus,
 xscale/.initial=1, % default xscale=1 (no scaling)
 yscale/.initial=0.6,% default yscale=0.6
 columns/.initial=2, % default of 2 columns
 rows/.initial=4 % default of 2 rows
}
newcommandAbacusValue[1]pgfkeysvalueof/tikz/abacus/#1

newcommandabacus[2][]tikzsetabacus, #1%
 begintikzpicture[
 xscale=AbacusValuexscale, 
 yscale=AbacusValueyscale
 ]
 foreach ypos in 1,...,AbacusValuerows 
 draw(0,2*ypos-1)--++(6,0);
 
 foreach xpos in 1,...,AbacusValuecolumns 
 draw(2*xpos, 0)--++(0,8);
 
 foreach row [count=ypos] in #2 
 foreach col [count=xpos] in row 
 draw (2*xpos-1,ypos) picabacus=col;
 
 
 endtikzpicture%


begindocument

 abacus2,1,1,1,3,3,2,,4,2,1
 qquad
 abacus1,,,,2,1

enddocument

The abacus command accepts an optional argument that can be used to change the default number of rows and columns and to change the x and y scaling. For example,

abacus[rows=2, columns=4,xscale=0.5,yscale=0.3]1,2,3,4,0,3,1

produces:

Here I draw the first picture. You can draw the other based on this.

documentclass[tikz]standalone
begindocument
begintikzpicture[x=1.5cm]
foreach i in 1,2 draw (i,-.2) -- (i,3.2);
foreach i in 0,1,2,3 draw (0,i) -- (3,i);
foreach x/y in 
 .5/0,
 .4/2,
 .6/2,
 .5/2.5
 fill (x,y) circle (2pt);
endtikzpicture
enddocument